A girl riding a bicycle with a speed of ` 5 m//s` to wards North direction, observes rain falling vertically down. If she increases her speed to `10 m//s`, rain appeared to meet her at ` 45^@` to the vertical . What is the speed of the rain ?

Explanation

To solve the problem, we will analyze the motion of the girl and the rain using vector components.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions:**
- The girl rides her bicycle at a speed of \(5 \, \text{m/s}\) towards the North (let's assume this is the positive y-direction).
- The rain is observed to fall vertically downwards, meaning it has no horizontal component in the frame of reference of the girl.

...